Revisiting User Mobility and Social Relationships in LBSNs:A Hypergraph Embedding Approach

Dingqi Yang, Bingqing Qu, Jie Yang and Philippe Cudre-Mauroux

University of Fribourg, Switzerland

{firstname.lastname}@unifr.ch

ABSTRACTLocation Based Social Networks (LBSNs) have been widely used as

a primary data source to study the impact of mobility and social re-

lationships on each other. Traditional approaches manually define

features to characterize users’ mobility homophily and social prox-

imity, and show that mobility and social features can help friendship

and location prediction tasks, respectively. However, these hand-

crafted features not only require tedious human efforts, but also

are difficult to generalize. In this paper, by revisiting user mobil-

ity and social relationships based on a large-scale LBSN dataset

collected over a long-term period, we propose LBSN2Vec, a hyper-

graph embedding approach designed specifically for LBSN data

for automatic feature learning. Specifically, LBSN data intrinsically

forms a hypergraph including both user-user edges (friendships)

and user-time-POI-semantic hyperedges (check-ins). Based on this

hypergraph, we first propose a random-walk-with-stay scheme to

jointly sample user check-ins and social relationships, and then

learn node embeddings from the sampled (hyper)edges by preserv-

ing n-wise node proximity (n = 2 or 4). Our evaluation results show

that LBSN2Vec both consistently and significantly outperforms the

state-of-the-art graph embedding methods on both friendship and

location prediction tasks, with an average improvement of 32.95%

and 25.32%, respectively. Moreover, using LBSN2Vec, we discover

the asymmetric impact of mobility and social relationships on pre-

dicting each other, which can serve as guidelines for future research

on friendship and location prediction in LBSNs.

CCS CONCEPTS• Information systems→ Data mining;Web applications;

KEYWORDSMobility, Social relationship, Location based social network, Link

prediction, Hypergraph, Embeddings

ACM Reference Format:Dingqi Yang, Bingqing Qu, Jie Yang and Philippe Cudre-Mauroux. 2019.

Revisiting User Mobility and Social Relationships in LBSNs: A Hypergraph

Embedding Approach. In Proceedings of the 2019 World Wide Web Conference(WWW ’19), May 13–17, 2019, San Francisco, CA, USA. ACM, New York, NY,

USA, 11 pages. https://doi.org/10.1145/3308558.3313635

This paper is published under the Creative Commons Attribution 4.0 International

(CC-BY 4.0) license. Authors reserve their rights to disseminate the work on their

personal and corporate Web sites with the appropriate attribution.

WWW ’19, May 13–17, 2019, San Francisco, CA, USA© 2019 IW3C2 (International World Wide Web Conference Committee), published

under Creative Commons CC-BY 4.0 License.

ACM ISBN 978-1-4503-6674-8/19/05.

https://doi.org/10.1145/3308558.3313635

1 INTRODUCTIONUnderstanding the correlation between human mobility and social

relationships is crucial for studying human dynamics, which is

also a key ingredient for friendship and location prediction tasks.

Location Based Social Networks (LBSNs), such as Foursquare, have

attracted millions of users and generated a considerable amount of

digital footprints from their daily life. As such, they have become

a primary data source to study the impact of mobility and social

relationships on each other. Specifically, in LBSNs, users can share

their real-time presences with their friends by checking-in at a

Point of Interest (POI), such as a restaurant or a gym. In addition

to such fine-grained and semantic user mobility information, the

social network of the corresponding users is also available.

Based on LBSN data, empirical analyses have shown various

socio-spatial properties of user activities, in particular the correla-

tion between usermobility characteristics (e.g., distance, co-location

rate, etc.) and social networks [8, 26–28, 42]. Based on these find-

ings, two main applications have been investigated, i.e., friendship

prediction (a.k.a. link prediction) and location prediction. Friendshipprediction aims at recommending social relationships that are likely

to be established in the future [28], while location prediction tries

to predict which POI a user will go to in a given context (e.g., at a

given time) [8]. Existing work has shown that considering the cor-

relation between user mobility and social relationships can improve

the performance of both friendship prediction [7, 25, 28, 32, 42]

and location prediction [8, 12, 18, 21, 25]. More precisely, these

approaches usually select a set of hand-crafted features either from

user mobility data or from the corresponding social network, and

then show the impact of one on the other. For example, social

features usually involve different network proximity metrics in-

cluding common neighbor, Adamic-Adar [1], Katz index [17], etc.

For mobility features, co-location rate is a widely used metric to

measure the homophily between two users in terms of mobility

traces; it has different variations such as normalized/unnormalized,

weighted/unweighted, spatial only/spatiotemporal [28, 33]. How-

ever, such a manual feature engineering process not only requires

tedious efforts from domain experts, but also shows less generaliz-

ability [13] (see Section 5.2.2 and 5.3.2 for more detail).

To overcome the limitation of hand-crafted features, automatic

feature learning (a.k.a. representation learning) has been proposed

[3]. When applied to networks or graphs, this paradigm is typically

known as graph (or network) embedding [5], which represents nodesof a graph in a low-dimensional vector space while preserving key

structural properties of the graph (e.g., topological proximity of the

nodes). Based on such embeddings, graph analysis tasks (e.g., link

prediction) can then be efficiently performed. As shown in Figure 1,

LBSN data in this context intrinsically form a hypergraph consist-

ing of four key data domains, i.e., spatial, temporal, semantic and

Figure 1: LBSN hypergraph containing both user mobilitydata (check-ins) and the corresponding social network. Afriendship is represented by an edge (a black dotted line)linking two user nodes. A check-in is represented by a hy-peredge (a colored thick line) linking four nodes, i.e., a user,an activity type, a time stamp and a POI.

social. This graph contains not only classical edges (i.e., friendships

between two users in the social network), but also hyperedges (i.e.,

check-ins linking four nodes, one from each domain, representing

a user’s presence at a POI at a specific time along with the semanticinformation about the user’s activity there).

However, existing graph embedding techniques cannot fully

grasp the complex data structure of LBSNs. First, most of the ex-

isting techniques were developed for classical graphs [6, 13, 22–

24, 29, 30], where an edge links two nodes only; the node embed-

dings are then learnt such that the pairwise node proximity is

preserved. However, preserving the pairwise node proximity can-

not fully capture the information from the check-in hyperedges.

Even though a hypergraph can be transformed into a classical graph

by breaking each hyperedge into multiple classical edges, such an

irreversible process causes a certain information loss, leading to

degraded performance of the learnt node embeddings on different

tasks (see Section 5.2 and 5.3 for more detail). Second, there are

also a few techniques studying the hypergraph embedding prob-

lem, but they either focus on a n-uniform hypergraph (where all

hyperedges contain a fixed number n of nodes) and thus capture

only fixed-n-wise node proximity [2, 31, 44], or learn from hyper-

edges for heterogeneous events only (hyperedges linking nodes

from different data domains) while ignoring edges within a data

domain [14]. However, as shown in Figure 1, the LBSN hypergraph

typically contains both classical friendship edges (n=2) within the

user domain and check-in hyperedges (n=4) across all four datadomains. Subsequently, these techniques cannot be directly applied

to the full LBSN hypergraph. Even though these techniques could

be applied on check-in hyperedges (n=4) only, ignoring social rela-

tionship indeed shows suboptimal performance for different tasks

(see Section 5.3 for more detail).

Against this background, we revisit human mobility and social

relationships in LBSNs in this paper and propose LBSN2Vec, a

hypergraph embedding approach designed specifically for LBSN

data with the special consideration of learning from both friendshipedges and check-in hyperedges at the same time. Our contributions

are three-fold:

• We collect a large-scale LBSN dataset over a long-term period and

conduct a rigorous empirical analysis to systematically reveal the

correlations between user check-ins and the corresponding social

network. Specifically, existing publicly available LBSN datasets

collected from Foursquare [12], Gowalla [8] or Brightkite [8],

usually contain check-in data from a set of users over a period of

time, and one snapshot of the corresponding user social network.

However, it is often unclear when the social network snapshot is

collected (e.g., before, during or after the check-in data collection

period); such chronological information is important in studying

the impact of user mobility and the corresponding social network

on each other. For example, to investigate the impact of social net-

works on user mobility patterns, only friendship formed before

a check-in should be considered. Therefore, our collected dataset

contains not only check-ins of a set of users over a two-year

period, but also two snapshots of the corresponding user social

network before and after the check-in data collection period, re-

spectively. Based on this dataset, we conduct a rigorous empirical

analysis of the impact of user mobility and social network on

each other, showing that each of the four data domains has a

clear impact on both friendship and location prediction tasks.

• We propose LBSN2Vec, a hypergraph embedding approach that

can efficiently learn node embeddings from both friendship edges

and check-in hyperedges in a LBSN hypergraph. To this end, we

first propose a random-walk-with-stay scheme to jointly sample

friendship and check-ins from a LBSN hypergraph. To balance

the impact of social relationships and user mobility on the learnt

node embeddings, we incorporate a tunable parameter to control

the portion of social relationships and check-ins in the learning

process in a probabilistic manner. Moreover, to learn node embed-

dings from hyperedges containing n nodes, we design LBSN2Vec

to preserve the n-wise node proximity by simultaneously opti-

mizing the proximity of the n nodes from a hyperedge. More

precisely, we first compute the best-fit-line of the n nodes in

cosine space, and then iteratively maximize the cosine similarity

between each node of the hyperedge and the best-fit-line.

• We conduct a thorough evaluation using our collected dataset.

Our results show that LBSN2Vec outperforms state-of-the-art

graph embedding techniques on both friendship and location pre-

diction tasks, with an average improvement of 32.95% and 25.32%,

respectively. Moreover, using LBSN2Vec, we discover the asym-

metric impact of user mobility and social networks on predicting

each other. We find that node embeddings learnt from 80% social

and 20% mobility data results in the best performance on the

friendship prediction task, while those learnt from 40% mobility

and 60% social data give the best performance on the location

prediction task. Such an observation can serve as guidelines for

future research on friendship and location prediction.

2 RELATEDWORK2.1 Human Mobility and Social RelationshipsThe interaction between human mobility and social relationships

has been widely studied using different data sources over the past

decade. In the earlier stage, call detail records have been often used

to study human mobility, social ties and link prediction problems

[8, 33]. Although these studies have revealed several interesting

properties of human mobility and social ties, such data shows two

obvious limitations. First, it contains only coarse-grained user loca-

tion information on a cell tower level. Second, it does not contain

actual information about social relationships between users; the

corresponding social network are usually built from the users’ com-

munication activities based on certain heuristics. For example, a

friendship is assumed between a pair of users when there is at least

one call between them [33] (or 10 calls in [8]).

The emergence of Location Based Social Networks provides a

novel opportunity to collect both large-scale user mobility data

(i.e., check-ins) and the corresponding social network. Using LBSN

data, positive correlations between social proximity and mobility

homophily have been universally found [8, 33]. However, existing

publicly available LBSN datasets (Foursquare [12], Gowalla [8] or

Brightkite [8]) mainly contain check-in data of a set of user over

a period of time, and one snapshot of the corresponding social

network without mentioning when the social network is collected1;

such chronological information is important in studying the impact

of user mobility and the corresponding social network on each

other. Therefore, in this paper, we collected our own dataset from

Foursquare, containing not only check-ins of a set of users over

about two years, but also two snapshots of the corresponding user

social network before and after the check-in data collection period,

respectively. Based on this dataset, we investigate the impact of user

check-ins and the corresponding social network on both friendship

and location prediction tasks.

First, friendship prediction is a classical problem in social net-

work analysis. It predicts the potential friendship between two users

based on the existing social network [19]. Using LBSN data, friend-

ship prediction approaches [7, 25, 28, 32, 42] often combine social

proximity and mobility pattern similarity to achieve better predic-

tion performance. For example, Scellato et al. [28] investigated a

set of hand-crafted features and showed that besides social proxim-

ity, mobility pattern similarity is also a strong indicator to predict

future friendship. Sadilek et al. [25] manually combined features

from social proximity, mobility similarity, and textual similarity

extracted from users’ Tweets for friendship prediction.

Second, location prediction is a typical problem in human mobil-

ity modeling, which predicts the location of a user under a certain

context based on the user’s historical mobility traces. Using LBSN

data, location prediction approaches incorporate social factors in

mobility models, showing an improved prediction performance

[8, 12, 18, 21, 25]. For example, Gao et al. [12] combined check-in

patterns and social ties by considering mobility similarity between

friends (co-location rate measured by cosine distance). Noulas et al.

[21] investigated the next place prediction problem and found that

besides a user’s own check-in history, a “social filtering” feature

serves as an important predictor for the user’s next location.

However, the manual feature engineering process adopted by

the existing work not only requires a lot of human efforts, but

also shows less generalizability [13]. In this paper, we propose

LBSN2Vec, a hypergraph embedding approach to automatically

learn the node embeddings from a LBSN hypergraph, based on

which both friendship and location prediction tasks can be effi-

ciently performed. Moreover, our LBSN2Vec can also model the

impact of user mobility and social relationships on each other.

1Note that one of the existing work [28] used multiple social network snapshots to

study the friendship prediction problem, but the dataset is not publicly accessible.

Moreover, it used hand-crafted features for friendship prediction.

2.2 Graph EmbeddingsMost existing graph embedding approaches focus on preserving

pairwise node proximity in a classical graph, which can be further

classified into two categories according to the embedding learning

process. First, factorization based approaches [6, 22, 24] measure

pairwise node proximity as a matrix using a certain network prox-

imity metric, such as common neighbor or Adamic-Adar, and then

factorize this proximity matrix using matrix factorization tech-

niques to learn the node embeddings. However, factorization-based

approaches have an intrinsic scalability limitation, due to the qua-

dratic complexity of matrix factorization algorithms [4]. Second,

graph-sampling based approaches [13, 16, 23, 29, 30] sample node

pairs (directly or via random walks) from a graph, and then design

specific models to learn node embeddings from the sampled node

pairs via stochastic optimization. These graph-sampling based ap-

proaches are able to scale up to large datasets, as their complexity

mainly depends on the number of the sampled node pairs.

Moreover, there are also a few approaches studying the hyper-

graph embedding problem, but they either focus on a n-uniform hy-

pergraph (where all hyperedges contain a fixed number n of nodes)

and thus capture only fixed-n-wise node proximity [2, 31, 44], or

learn from hyperedges for heterogeneous event only (hyperedges

linking nodes from different data domains) while ignoring edges

within a data domain [14]. However, in this paper, as shown in

Figure 1, our LBSN hypergraph contains both classical friendship

edges (n=2) within the user domain and check-in hyperedges (n=4)across all four data domains. Subsequently, these techniques cannot

be directly applied to our full LBSN hypergraph. Even though they

can be applied only on the check-in hyperedges (n=4) for example,

ignoring social relationships indeed yields suboptimal performance

(see Section 5.3 for more detail). Against this background, we pro-

pose LBSN2Vec to efficiently learn node embeddings from the LBSN

hypergraph by preserving the n-wise node proximity encoded by

the hyperedges (where {n ∈ Z+ |n ≥ 2}).

We also note that there are several works exploited embedding

techniques on LBSN data, but they mostly focus on check-in data

only (without using social networks) for specific tasks, such as

semantic annotation of POIs [34], POI recommendation [10, 35],

next visitor prediction [11]. Only one recent work [43] used both

social network and check-in data on LBSNs for learning embeddings

for users and POIs, but it overlooks the temporal and semantic

information of check-ins. Moreover, none of these works have

investigated the correlations between human mobility and social

relationships.

3 EMPIRICAL DATA ANALYSISIn this section, we first present our dataset collection and its main

characteristics, followed by empirical studies on the impact of mo-

bility on the social network and vice versa.

3.1 Dataset Collection and CharacteristicsOur collected Foursquare dataset contains a set of global-scale

check-ins over about two years (from Apr. 2012 to Jan. 2014), and

two snapshots of the corresponding user social network before

(in Mar. 2012) and after (in May 2014) the check-in data collection

period. The check-in data is collected from Foursquare, which has

been widely used as a key data source for studying location based

services [39–41]. Specifically, as Foursquare user’s check-ins can

only be accessed from the user’s social circle, they are not publicly

available. However, many Foursquare users choose to also share

their check-ins via Twitter [37]. Therefore, we collect global-scale

check-in data by searching Foursquare-tagged Tweets from Twitter

Public Streams for about two years (from Apr. 2012 to Jan. 2014). We

keep only active users (who have performed at least one check-in

per month). User social relationships are collected from Twitter, as

Foursquare friendship information is not publicly available. Specifi-

cally, for each user, we first crawl her “followers” (users who follow

this user) and “friends” (users who are followed by this user) on

Twitter. We then build the corresponding user social network by

keeping only reciprocal relationships [38], i.e., a friendship is as-

sumed between two users if the two users follow each other. We

collect two snapshots of such social network before (in Mar. 2012)

and after (in May 2014) the check-in data collection period. After

filtering out isolated users (i.e., users without social links), we keep

only users appearing in both social network snapshots. In summary,

our dataset contains 22,809,624 check-ins from 114,324 users on

3,820,891 POIs, and two snapshots of the corresponding user social

network before (363,704 friendships) and after (607,333 friendships)

the check-in data collection period. Our dataset is available here2.

Figure 2 shows a series of visualizations of our dataset in all four

data domains. For the social domain, Figure 2(a) shows the old social

network with a color bar indicating user node degrees. We observe

only a few nodes with high degrees, while most of the nodes are

of low degrees, which suggests a heavy-tailed distribution. We do

not show the new social network as its plot shows a very similar

visual structure as the old one. For the semantic domain, Figure

2(b) shows the categorical distribution of check-in POIs, where we

have a total number of 453 POI categories (i.e., activity categories).

We find that daily-routine related activities including “Shopping

Mall”, “Home”, “Train station” and “Office”, are among the most

frequent categories. For the temporal domain, Figure 2(c) plots

the weekly temporal traffic pattern of check-ins over 168 hours

in a week. We observe not only a clear daily periodicity, but also

the difference between weekday (with three peaks in the morning,

afternoon and evening, respectively) and weekend (with only one

flat peak during the daytime). Based on this observation, we define

the time granularity (time slot) as 168 hours (corresponding to a

full week) for this study. For the spatial domain, we visualize the

GPS coordinates of POIs on a world map in Figure 2(d). We observe

POIs from our dataset spread all over the world, with urban areas

concentrating many of the POIs.

In addition, we take a closer look at the dataset from a statistical

point of view. Figure 3 shows the Complementary Cumulative

Distribution Function (CCDF) of different statistics of the collected

dataset. First, we observe a heavy-tailed distribution of all statistics,

including user node degrees in Figure 3(a), number of check-ins

per user in Figure 3(b), number of check-ins per POI in Figure 3(c),

and check-in distance from home location [8] in Figure 3(d), which

implies a large variance in these statistics. For example, the heavy-

tailed distribution of check-in distance from home shows that 70%

of the check-ins happen less than 100km from home. Moreover, we

2https://sites.google.com/site/yangdingqi/home/foursquare-dataset

(a) Social network visualization (b) Check-in POI categories

(c) Temporal traffic of check-ins (d) World map of check-in POIs

Figure 2: Visualization of the collected LBSN data

(a) Node (user) degree (b) Number of check-ins per user

(c) Number of check-ins per POI (d) Check-in distance from home

Figure 3: Complementary Cumulative Distribution Func-tion (CCDF) of various statistics in the collected dataset

observe that the user social network significantly expanded over

the two years in Figure 3(a); the average degree increases from 3.38

to 5.56. In the following, we study the impact of mobility on the

corresponding social network and vice versa.

3.2 Impact of Mobility on New FriendshipsWe investigate the probability of establishing a new friendship

(appearing in the new but not in the old social network), w.r.t.

various social andmobility factors in the old social network and user

check-in data. Figure 4 shows the results. On one hand, the existing

social network is unsurprisingly the primary predictor of future

friendships [19]. Figure 4(a) shows the impact of the social proximity

between users in the old social network. We observe that pairs of

users at closer social distance are more likely to establish a new

friendship. On the other hand, we use the following three metrics to

measure the similarity of user mobility patterns on spatial, temporal

and semantic domains, respectively.

• Co-location [33]: By modeling each user’s check-in data as a

vector, where each element in the vector indicates whether the

user has checked-in at the corresponding POI or not, we measure

the Jaccard distance between two users’ check-in vectors.

(a) Social proximity (b) Co-location

(c) Temporal activeness (d) Activity semantics

Figure 4: Impact of social and mobility factors on establish-ing new friendships

• Temporal activeness [36]: We model a user’s check-in data by

a vector encoding the user’s check-in frequency over 168 hours

in a week (as shown in Figure 2(c)), and then measure the Cosine

distance between two users’ temporal activeness vectors.

• Activity semantics [40]: We model a user’s check-in data by a

vector encoding the categorical distribution of a user’s checked

POIs (i.e., the distribution on POI categories as shown in Figure

2(b)), and then measure the Cosine distance between two users’

activity semantic vectors.

Figure 4(b), 4(c) and 4(d) show the impact of the above three metrics

on the probability of establishing a new friendship, respectively.

We consistently observe that pairs of users with similar mobility

patterns (measured in any of the three domains) are more likely to

establish a new friendship in the future. These observations suggestthat all four data domains should be considered in the friendshipprediction task. Note that we only use these metrics to investigatewhich data domains should be considered in the LBSN2Vec, yet noneof these metric is required in our node embedding learning process.

3.3 Impact of Friendship on MobilityWe investigate the similarity of user mobility patterns w.r.t. the

social proximity between users in the old social network. Figure

5 shows the results. We consistently observe that pairs of users at

closer social proximity are more likely to exhibit similar mobility

patternsmeasured by all the threemetrics, i.e., co-location, temporal

activeness and activity semantics. Such an observation suggests

that social relationships can be used to help predict user mobility.

In addition, we also investigate the impact of a user’s historical

mobility traces on her future mobility. To achieve this goal, we

chronologically split our check-in data (of about two years) into

two parts of the same duration, i.e., a first-year check-in dataset

and a second-year check-in dataset. Subsequently, we compute the

similarity between each user’s mobility patterns in the first and the

second year, measured by the aforementioned three metrics. The

average similarity is 0.12, 0.32 and 0.38 for co-location, temporal

activeness and activity semantics, respectively. Compared to the

mobility similarity between friends (the distance in the old social

network is 1 in Figure 5, i.e., 0.04, 0.16 and 0.29 for co-location,

(a) Co-location (b) Temporal activeness (c) Activity semantics

Figure 5: Impact of social proximity on mobility factors

temporal activeness and activity semantics, respectively), we find

that a user’s own mobility history is indeed the primary predictor

of her future mobility. Subsequently, we also consider all four datadomains in our LBSN2Vec for node embedding learning in the locationprediction task. We note again that as an automatic feature learningapproach, LBSN2Vec does not involve any of the metrics defined inthis section.

4 LBSN2VECIn this paper, we adopt a graph-sampling based embedding para-

digm due to its intrinsic scalability advantage as discussed in Section

2.2. Specifically, our method first uses a novel random-walk-with-

stay scheme to jointly sample friendships and check-in hyperedges3

from the LBSN hypergraph, and then learn node embeddings from

these hyperedges to preserve the n-wise node proximity by optimiz-

ing the proximity of the n nodes from a hyperedge simultaneously.

4.1 RandomWalk with StayAs shown in Figure 1, the LBSN hypergraph consists of four data

domains, i.e., spatial, temporal, semantic and social domain, and

two types of edges, i.e., classical edges (friendships) linking pairs of

user nodes and hyperedges (check-ins) linking four nodes, one from

each domain. To sample (hyper)edges from this graph, we propose

a random-walk-with-stay scheme to jointly sample friendship and

check-in hyperedges. As shown in Figure 6, our random-walk-

with-stay scheme performs classical random walk only on user

nodes based on their friendships in social domain, while for each

encountered user node, it stays on the user node to sample a set of

hyperedges (check-ins) from the corresponding user. Subsequently,

in the node embedding learning process, when iterating over each

user node in a random walk sequence, we not only learn from two

user nodes appearing with a context window of length k , but alsostay on the corresponding user node to learn from its check-in

hyperedges. In other words, the node embedding learning process

alternates between these two types of edges (i.e., friendships and

check-ins). To perform classical random walk on the user social

network, we use the same strategy as used by existing works [13,

23], generating r walks of length l rooted on each user node.

Moreover, in order to balance the impact of friendships and

check-ins on the learnt node embeddings, LBSN2Vec incorporates a

tunable parameter α to control the portion of learning edges of each

type. Specifically, for a given context window of length k , we iterateover 2k contexts for each user node vi (k context nodes on each

side), resulting in 2k pairs of user nodes. Figure 6 shows an example

3A classical edge linking two user nodes (i.e., a friendship) can be regarded as a special

case of a hyperedge consisting of only two nodes. In the following, besides the check-in

hyperedges, we also use the term “hyperedge” to describe a pair of user nodes.

Figure 6: Random walk with stay on the LBSN hypergraph

(a) Euclidean space with perpendic-

ular distance

(b) Cosine space with cosine dis-

tance

Figure 7: Examples of the best-fit-lines in Euclidean and Co-sine spaces. The given data points are represented in blue,while the corresponding best-fit-line is shown in red.

where the context window size is 2. Subsequently, we sample for

each user node the same number (2k) of check-ins. Afterwards, welearn from each check-in with a probability α , while from each user

node pair with a probability (1 − α), where α and (1 − α) actuallyspecify the impact of mobility and social network on the learnt node

embeddings, respectively. In summary, for each nodevi in a randomwalk, the expected total number of learnt edges is 2k , includingan expected number 2kα of check-ins and an expected number of

2k(1−α) of user node pairs. A small value of α gives less importance

on the check-in data and more on social network, and vice versa.

In the following, we present our node embedding learning process

preserving the n-wise node proximity for hyperedges.

4.2 Learning from HyperedgesWe design a novel embedding model in LBSN2Vec to preserve the

n-wise node proximity by maximizing the similarity between the

nodes of a hypergraph and their best-fit-line under cosine similarity.

Specifically, our embedding model needs to learn from hyperedges

linking either 2 or 4 nodes. We further generalize this problem to

hyperedges containing n nodes, where {n ∈ Z+ |n ≥ 2}, and design

our model that can preserve the n-wise node proximity from those

hyperedges. As the hyperedges are actually sampled and learnt

one by one from a random walk, we focus in the following on the

learning process for one hyperedge.

To learn node embeddings from a hyperedge containing n nodes,

we borrow the idea of best-fit-line (a.k.a. line of best fit) from linear

regression [15]. In general, a best-fit-line is a straight line that is the

best approximation of the given data points. Figure 7(a) illustrates

an example of the best-fit-line in Euclidean space that minimizes the

sum of perpendicular distances, which can be computed via linear

least squares. In this paper, following existing graph embedding

techniques [6, 13, 22–24, 29, 30], we learn node embeddings in

cosine space where the proximity between nodes are measured

using cosine similarity (or dot product of the normalized embedding

vectors). Moreover, we show below that the computation of the best-

fit-line in cosine space can be significantly simplified. Formally, the

best-fit-line of a set of node embeddings is the vector that minimizes

the sum of cosine distances between each node embedding and

the best-fit-line, as shown in Figure 7(b), which can be efficiently

computed using Proposition 1:

Proposition 1. For a set of nodes (in a hyperedge) {vi |i = 1, 2, . . . ,n},the corresponding best-fit-line can be computed as ®vb =

∑ni=1

®vi∥ ®vi ∥

,where ®vi refer to the embedding vector of node vi .

Proof. The problem of finding the best-fit-line ®vb that mini-

mizes the sum of cosine distances can be formulated as follows:

argmin

®vb

n∑i=1

(1 − cos(®vi , ®vb )) (1)

As minimizing the cosine distance is equivalent to maximizing the

cosine similarity, we obtain:

argmax

®vb

n∑i=1

cos(®vi , ®vb ) = argmax

®vb(

n∑i=1

®vi∥ ®vi ∥

) ·®vb

∥ ®vb ∥(2)

Let ®va =∑ni=1

®vi∥ ®vi ∥

, we obtain:

argmax

®vb®va ·

®vb∥ ®vb ∥

= argmax

®vb∥ ®va ∥

®va · ®vb∥ ®va ∥∥ ®vb ∥

(3)

where ∥ ®va ∥ is a constant for the given set of node. Subsequently, theabove function actually maximizes the cosine similarity between ®vaand ®vb . As the maximal cosine similarity between any two vectors

is achieved if and only if the two vectors have the same orientation,

the best-fit-line ®vb can be simply computed as:

®vb = ®va =n∑i=1

®vi∥ ®vi ∥

(4)

This completes the proof. □

Therefore, we optimize the n-wise node proximity by iteratively

maximizing the cosine similarity between each node embedding ®vifrom the hyperedge {vi |i = 1, 2, . . . ,n} and the best-fit-line ®vb .

Θ =n∑i=1

cos(®vi , ®vb ) (5)

Note that here we keep using cosine similarity rather than dot

product (which is widely used by existing techniques), because

cosine similarity is not affected by the norm of the input vectors,

in particular when the norm of the best-fit-line ®vb computed by Eq.

4 has a large variance due to the variant number of the nodes n in

different hyperedges (n = 2 or 4 in our LBSN hypergraph).

In addition, similar to other graph-sampling based embedding

techniques [13, 23, 29, 30], we also adopt a negative sampling

technique to maximize the cosine distance between each negative

sample node vN and the best-fit-line vector ®vb . Finally, we wantto maximize the following objective function for one hyperedge

{vi |i = 1, 2, . . . ,n}:

Θ =n∑i=1

(cos(®vi , ®vb ) + γ · EvN [1 − cos(®vN · ®vb )]

)(6)

where γ ∈ Z+ is the number of negative samples. Note that as the

LBSN hypergraph contains four data domains, the negative samples

for a node vi from one data domain are uniformly sampled from

the nodes in the same data domain.

Table 1: Statistics of the selected datasetsDataset NYC TKY IST JK KL SP#User 4,024 7,232 10,367 6,395 6,432 3,954

#POI 3,628 10,856 12,693 8,826 10,817 6,286

#Check-ins 105,961 699,324 908,162 378,559 526,405 249,839

#Friendships(Before) 8,723 37,480 21,354 11,207 16,161 9,655

#Friendships(After) 10,545 51,704 36,007 16,950 31,178 14,402

The above objective function can be optimized using stochastic

gradient descent techniques. Note that we only need to update ®viand ®vN when learning from one specific hyperedge, as the best-fit-

line vector ®vb is fixed. The gradient of the above objective function

with respect to ®vi and ®vN is computed as follows:

∂Θ

∂ ®vi=

®vb∥ ®vi ∥∥ ®vb ∥

−®vi cos(®vi , ®vb )

∥ ®vi ∥2(7)

∂Θ

∂ ®vN= −

®vb∥ ®vN ∥∥ ®vb ∥

+®vN cos(®vN , ®vb )

∥ ®vN ∥2(8)

In summary, to learn node embeddings from the LBSN hyper-

graph, we first sample a set of friendship and check-in hyperedges

using our proposed random-walk-with-stay scheme, and then it-

eratively learn from each hyperedge {vi |i = 1, 2, . . . ,n} (n = 2

for friendships or n = 4 for check-ins) by optimizing the objective

function in Eq. 6.

5 EXPERIMENTSIn this section, we evaluate LBSN2Vec on both friendship and loca-

tion prediction tasks using our collected dataset. In the following,

we first present our experimental setup, and then discuss the results

on individual tasks, followed by a parameter sensitivity study.

5.1 Experimental Setup5.1.1 Dataset. In this study, we investigate the friendship and lo-

cation prediction at an urban scale. Without loss of generality, we

select six cities with a large number of check-ins, while also con-

sidering the cultural diversity of the selected cities: New York City

(NYC), Tokyo (TKY), Istanbul (IST), Jakarta (JK), Kuala Lumpur

(KL), Sao Paulo (SP). We select users in the largest connected com-

ponents of the old social network. Table 1 summarizes the statistics

of the selected datasets.

5.1.2 Baselines. As our objective is automatic feature learning

from LBSN data, we compare our method to the following state-of-

the-art graph embedding methods.

• DeepWalk [23] first generates node sequences using random

walks, and then feeds them to a SkipGram model to output node

embeddings. We set the walk length to l = 80, the number of

walks per node to r = 10, and the context window size for the

SkipGram model to k = 10.

• Node2vec [13] extends DeepWalk by introducing a parameter-

ized random walk method to balance the breadth-first search

(return parameter p) and depth-first search (in-out parameter q)strategies, to capture richer graph structures. Following the sug-

gestions in the original paper, we tune p and q with a grid search

over p,q ∈ {0.25, 0.05, 1, 2, 4}. We keep the other parameters

l , r ,k similar as for DeepWalk.

• LINE [29] explicitly defines 1st and 2nd-order node proximity

to learn graph embeddings. Specifically, to learn node embed-

dings of dimension d , it first learns two sets of d/2-dimension

node embeddings preserving 1st and 2nd-order node proximity,

respectively, and then concatenates them together.

• HEBE [14] learns node embeddings for heterogeneous event datausing tensor modeling, where a heterogeneous hyperedge links

nodes across different domains only and is modeled as an entry

in an high-order tensor.

• DHNE [31] uses a deep autoencoder to learning node embed-

dings from a n-uniform heterogeneous hypergraph. It preservesboth fixed-n-wise node proximity encoded by the hyperedges,

and also the high-order node proximity defined as the similarity

between nodes’ neighborhood structures.

As the baseline methods cannot directly take the LBSN hyper-

graph as input graph, we propose three settings to adapt our LBSN

hypergraph:

• (S): Only the Social network is considered by keeping nodes

in user domain and their friendship edges. This setting can be

applied to classical graph embedding techniques (preserving

pairwise node proximity), i.e., DeepWalk, Node2vec and LINE.• (M): Only the user Mobility is considered by keeping check-in

hyperedges only. This setting can be directly applied to HEBEand DHNE which can take heterogeneous check-in hyperedges

as input. In addition, we also apply this setting to DeepWalk,Node2vec and LINE by breaking each check-in hyperedge into

multiple classical edges linking each pair of nodes in the hyper-

edge (e.g., user-time, time-POI, etc.).

• (S&M): The full LBSN hypergraph containing both friendship

edges and check-in hyperedges. This setting can be directly ap-

plied to our LBSN2Vec. In addition, we also apply this setting

to DeepWalk, Node2vec and LINE by mapping the full LBSN

hypergraph onto a classical graph by keeping friendship edges

and also breaking each check-in hyperedge into multiple classical

edges linking each pair of nodes in the hyperedge (same as for

the M setting).

For our method LBSN2Vec, we keep the same random walk

parameters as for DeepWalk and Node2vec (l = 80, r = 10, k = 10),

and tune the parameter α within [0, 1] with a step of 0.1 to balance

the impact of social and mobility on the learnt embeddings (more

discussion on this point in Section 5.4). The dimension of the node

embeddings d is set to 128 and the number of negative samples

γ is set to 10 for all methods in all experiments, if not specified

otherwise. The implementation of LBSN2Vec is available here4.

5.2 Friendship (link) PredictionFriendship prediction recommends friendships that will probably

be established in the future. In this study, we advocate an unsu-

pervised friendship prediction approach [19, 20] which generates a

ranking list of potential links between pairs of users. Specifically,

after learning the node embeddings based on the old social network

and check-ins, we rank pairs of user nodes (not being friends in the

old social network) according to the cosine similarity between their

embeddings. We then evaluate the obtained ranking list against

4https://github.com/eXascaleInfolab/LBSN2Vec

Figure 8: Friendship prediction performance comparisonwith other graph embedding methods. Each row shows theresults on one dataset, while each column shows the resultsfor one evaluation metric.

the new friendships (appearing in the new but not in the old so-

cial network), using three common metrics for evaluating ranked

results, i.e., precision, recall, and F1-score on the top K predicted

friendships [19]. As the number of candidate pairs of nodes is too

large, we randomly sample 1% pairs of nodes for the evaluation,

and report the average results from 10 repeated trials.

5.2.1 Comparison with other graph embedding methods. We report

only the methods including social networks (setting S and S&M),

since the social proximity in the old social network is indeed the pri-

mary predictor of new friendships (see Figure 4(a) for more details)

and since we found out that methods using setting M perform very

poorly on this task. Figure 8 shows the results. We clearly observe

that LBSN2Vec achieves the highest precision, recall and F1-score

in general. For example, we find that LBSN2Vec outperforms all

baselines with an average improvement of 32.95% on precision@10

over the best baselines across different datasets.

One interesting observation is that for all the baseline methods,

considering both the social network and mobility (setting S&M)

leads to worse performance than considering only the social net-

work (setting S). In other words, for all baseline methods, adding

user mobility information decreases the friendship prediction per-

formance. Despite the counter-intuitive nature of this observation,

Figure 9: Friendship prediction performance comparisonwith hand-crafted features.

we notice that these methods fail to balance the impact of social

network and user mobility on the learnt node embeddings. More

precisely, as the number of check-ins is usually much larger than the

number of friendships, the S&M graph is dominated by the edges

representing check-ins. When the baseline methods uniformly sam-

ple edges from the S&M graph to learn the node embeddings, the

sampled edges are also dominated by the edges representing check-

ins, which naturally imposes a strong impact of mobility on the

learnt node embeddings. In fact, as mentioned for Figure 4(a) in

Section 3.2, the social proximity in the old social network is indeed

the primary predictor of new friendships (our parameter sensitivity

study in Section 5.4 also verifies this point below). Therefore, the

learnt node embeddings from those baseline methods (with setting

M) result in the worst performance in friendship prediction.

5.2.2 Comparison with hand-crafted features. We compare LBSN2Vec

with the following hand-crafted features suggested specifically for

the friendship prediction task by previous work:

• Katz Index (Katz) [17] is the best performing social feature sug-

gested by [33]. It is defined as the weighted sum of all paths

between two users on the social networks, where the weight

of a path decays exponentially with its length. Katz(u,v) =∑∞l=1 β

l · |pathlu,v |, where pathlu,v is the set of all paths with

length l from u to v . We set β = 0.05 as suggested by [33].

Figure 10: Location prediction performance comparisonwith other graph embedding methods on all test check-ins

• Adamic-Adar (AA) [1] is the best performing social feature sug-

gested by [28]. It is the weighted sum of the common neighbors

between two users. The weight for each common neighbor is the

inverse logarithm of its degree.AA(u,v) =∑z∈Γ(u)∩Γ(v)

1

log |Γ(z) | ,

where Γ(u) is the set of neighbors (friends) of user u.• Spatial Cosine Similarity (SCos) is the best performing mobility

feature suggested by [33]. It is defined as the cosine similarity

between two users’ check-in vectorsCu andCv , i.e., cos(Cu ,Cv ),where each element Cu (i) is u’s check-in count on POI i .

• Minimum place entropy (Min_ent) is the best performing mo-

bility feature suggested by [28]. It is defined as the minimum

POI entropy (i.e., the entropy of the empirical distribution of

check-ins at that POI over users) that two users share.

As combining social and mobility features is suggested by both

[33] and [28], we consider all the combinations of the above so-

cial and mobility features, including Katz+SCos, Katz+Min_ent,AA+Min_ent and AA+SCos. Specifically, as these features havedifferent value ranges, we first generate one ranking list of pre-

dicted friendships using each feature, and then merge two ranking

lists using reciprocal rank fusion [9], which is a popular rank fu-

sion method widely used in information retrieval research. Figure 9

shows the results. We clearly observe that LBSN2Vec outperforms

the hand-crafted feature based methods in general, and we find

an average improvement of 66.44% on precision@10 over the best

baselines across different datasets. Moreover, We observe that none

of the four combinations can consistently outperform others over

all datasets, which shows the generalization limitations of the hand

crafted features across different datasets.

5.3 Location PredictionLocation prediction tries to predict the POI a user will be located

in at a given time slot. To implement this task, we chronologically

split our check-in data into two parts, i.e., the first 80% for training

and the last 20% for testing. We then learn node embeddings from

the old social network and the training check-in data. Based on the

learnt node embeddings, for each test check-in (containing a user,

a time slot and a POI), we rank all POIs according to the sum of

two cosine similarities: one between the node embeddings of the

user and a candidate POI, and one between the node embeddings

of the time slot and the candidate POI. For DHNE, we use its own

similarity function for ranking POIs. Subsequently, we evaluate the

obtained POI ranking list against the actual POI in the test check-in.

We report the average accuracy@10 over all test check-ins.

5.3.1 Comparison with other graph embedding methods. Figure 10shows the results. Note that only settings involving check-in data

(setting M and S&M) are eligible for this task. First, we observe

Figure 11: Location prediction performance comparisonwith other graph embedding methods on new POIs.

that our LBSN2Vec consistently outperforms all baselines with an

average improvement of 25.32% on accuracy@10 over the best base-

lines across different datasets. Second, we see that using setting

M, hypergraph embedding techniques (HEBE and DHNE) achieve

better performance than classical graph embedding techniques (pre-

serving only pairwise node proximity) in general, as the irreversible

process of transforming a hypergraph into a classical graph causes

a certain information loss, leading to degraded performance for

the location prediction task. Finally, similar to friendship predic-

tion, we also find that for all baselines considering both the social

network and mobility (setting S&M) results in worse performance

than considering mobility (M) only, as the baseline methods fail

to balance the impact of the social network and user mobility on

the learnt node embeddings. However, the performance drop is

relatively smaller than that of the friendship prediction task, as the

S&M graph is actually dominated by the edges from the check-ins.

In addition, we also investigate the prediction performance on

new POIs only, i.e., the POIs appearing in the test check-ins but

not in the training check-ins. As historical (training) check-ins of a

user are represented as hyperedges in the LBSN hypergraph, check-

ins with new POIs actually refer to the non-observed hyperedges

that will be established in the future. Therefore, this task is more

difficult as it requires to explore the LBSN hypergraph structure

for predicting potential hyperedges, rather than preserving the

observed hypergraph structure. Figure 11 shows the accuracy@10

for only new POIs. We observe that compared to the case of all

testing POIs, the improvement of our method over the baselines are

much higher for new POIs. We observe an average improvement of

50.17% on accuracy@10 in the case of only new POIs, compared to

an average improvement of 25.32% in the case of all test POIs. Such

an observation demonstrates the strong advantage of LBSN2Vec in

predicting potential check-in hyperedges.

5.3.2 Comparison with hand-crafted features. We also compare

LBSN2Vecwith the following hand-crafted features suggested specif-

ically for the location prediction task by previous work:

• Most Frequent POI (MFP) is the best performing mobility feature

suggested by [21]. For each user, it ranks a POI according to the

number of the user’s check-ins at that POI in the training dataset.

• Most Frequent Time (MFT) is suggested by [40]. For each user

and each time slot, it ranks a POI according to the number of

user’s check-ins at that POI and at that time slot in the training

dataset.

• Social Filtering (SF) is suggested by [21]. For a user u and his

friends Γ(u), it ranks a POI according to the total number of

check-ins that any friend v (v ∈ Γ(u)) of the user has performed

at the POI.

Figure 12: Location prediction performance comparisonwith hand-craft features on new POIs.

• Weighted Social Filtering (WSF) is suggested by [12]. It is aweightedversion of SF, where each friend’s check-in count is weighed by

the mobility similarity between the user and that friend. It ranks

a POI according to the weighted sum of the number of check-ins

that any her friend v , v ∈ Γ(u), has performed at that POI.

Similar to the friendship prediction task, we combine the results

from mobility and social features using reciprocal rank fusion [9],

resulting in four combinations (i.e.,MFP+SF,MFP+WSF,MFT+SF,MFT+WSF). Figure 12 shows the results on new POIs. We observe

that our method outperforms the hand-crafted feature in most cases

(expect for the TKY dataset where MFP+SF is slightly better). In

addition, similar to the case of friendship prediction, none of the

four combinations consistently achieve higher accuracy than others,

showing the generalization limitations of hand-crafted features.

5.4 Parameter Sensitivity Study5.4.1 Balance trade-off between social and mobility with α . We

investigate the trade-off between the social network and mobility

patterns on both the friendship and location prediction tasks by

varying α within [0, 1] with a step of 0.1. A small value of α gives

more importance on the social network and less on check-in data

when learning node embeddings, and vice versa. Figure 13 shows

the results. We see a clear trade-off between the social network and

mobility on all datasets. On one hand, when increasing α from 0, the

friendship prediction performance slightly increases and reaches

its peak around α = 0.2, as considering mobility factors indeed

helps the friendship prediction task. When further increasing α ,the performance start to decrease, and we observe a sharp drop

around α = 0.5. On the other hand, when decreasing α from 1, the

location prediction performance slightly increases and reaches its

peak around α = 0.6 in most cases (expect for KL dataset), meaning

that considering social factors can also help the location prediction

task. When further decreasing α , the performance start to decrease,

but we observe a sharp drop around α = 0.1 or 0.2. Interestingly,

by comparing the values of α giving the peak performance in the

two tasks, we find the asymmetric impact of social and mobility

factor on each other, which seems to be universal across different

datasets. More precisely, combining 80% social with 20% mobility

data results in the best performance for the friendship prediction

task, while combining 40% mobility with 60% social data gives the

best performance for the location prediction task.

5.4.2 Dimension of the node embeddings d . We study the impact

of the node embedding dimension d on both the friendship and lo-

cation prediction tasks, as well as on the node embedding learning

time on a commodity PC (Intel Core i7-4770HQ@2.20GHz, 16GB

RAM, Mac OS X). Figure 14 shows the results on the NYC dataset.

Figure 13: Impact of α on friendship and location predictiontasks

Figure 14: Impact of the node embedding dimensiond on theNYC dataset

First, we see that a small d leads to poor performance on both tasks,

as the resulting node embeddings cannot capture sufficient informa-

tion from the LBSN hypergraph. Then, the performance increases

with increasing d , and converges around d = 128, meaning that

further increasing d leads to little performance improvement. In

contrast, the embedding learning time always increases with in-

creasing d , as a higher dimension of node embedding requires more

computation in the node embedding learning process. Therefore,

we select d = 128 for all previous experiments.

6 CONCLUSIONIn this paper, by revisiting user mobility and social relationships

in LBSNs, we propose LBSN2Vec, a hypergraph embedding ap-

proach designed specifically for LBSN data. First, by collecting a

large-scale LBSN dataset over a long-term period, we conducted a

rigorous empirical analysis to reveal the correlations between user

check-ins and the corresponding social network. Then, we designed

LBSN2Vec to learn node embeddings from the LBSN hypergraph.

It performs random-walk-with-stay to jointly sample user mobil-

ity patterns and social relationships from the LBSN hypergraph,

and then learns node embeddings from the sampled hyperedges

by preserving the n-wise node proximity captured by a hyperedge

containing n nodes. Using the collected LBSN data, our extensive

evaluation shows that LBSN2Vec consistently outperforms the state-

of-the-art graph embedding methods on both the friendship and

location prediction tasks, with an average improvement of 32.95%

and 25.32%, respectively. Moreover, using LBSN2Vec, we discover

the asymmetric impact of mobility and social relationships on pre-

dicting each other, which can serve as guidelines for future research

on friendship and location prediction in LBSNs.

In the future, we plan to extend LBSN2Vec by further considering

sequential patterns in the user check-in data.

ACKNOWLEDGMENTSThis project has received funding from the European Research

Council (ERC) under the European Union’s Horizon 2020 research

and innovation programme (grant agreement 683253/GraphInt).

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